#187938

order	512
nilpotency class	3
isoclinism	isoclinic class of #187934
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#62818	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(2,7,10,19)(4,15)(8,18,20,31)(9,17,21,29)(13,23)(16,25)(22,28,32,30)(26,27);(1,2)(3,8)(4,7)(5,10)(6,9)(11,20)(12,21)(13,18)(14,22)(15,19)(16,17)(23,31)(24,32)(25,29)(26,30)(27,28);(3,11)(4,16)(7,17)(8,20)(13,26)(14,24)(15,25)(18,30)(19,29)(22,32)(23,27)(28,31);(1,3,12,24)(2,8,21,32)(4,13,25,26)(5,11,6,14)(7,18,29,30)(9,22,10,20)(15,23,16,27)(17,28,19,31)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,b^a=be,c^a=cf,e^a=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234516, #72234547, #72234608, …
maximal normal subgroup of	#72234516, #72234547, #72234608, …
maximal quotient of	#72313592

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95134);